What is the Grothendieck group of finitely generated $R[G]$-modules?

Question

Let $R$ be a ring with unity, $G$ a finite group and $R[G]$ the group ring. What is the definition of the Grothendieck group of finitely generated $R[G]$-modules?

How is this connected to the Grothendieck group of a ring?

Many thanks in advance.

Answer 1

There is the notion of the Grothendieck group of the category of finitely generated $R[G]$-modules (sometimes this is just named the Grothendieck group of finitely generated $R[G]$-modules).
The Grothendieck group is defined in the same way as before as the abelian group with one generator $[M]$ for each isomorphism class of objects of the category, and one relation $[A]-[B]+[C] = 0$ for each exact sequence $A\hookrightarrow B\twoheadrightarrow C$.